It seems my previous comment has been deleted, that's a bit irritating, but let's try to be more precise. The definition of an imaginary number is slightly wrong, and math doesn't suffer inaccuracy. i is defined as a solution of the equation x²+1=0 (the other one being of course -i), thus imaginary numbers are numbers whose square is a negative number. I know it seems to be the same, but the function square root is not well-defined on negative number, and cannot be in an analytic way. That function maps the positive real numbers on themselves. To say things otherwise, the square root of -1 (for example) is not well-defined since it could be i or -i indifferently.
The things people take offense to on this site continue to baffle me. It's like y'all need a 3 paragraph clue to get all of the little pedantry just right so you can properly answer the question. In the world of quiz writing, brevity is king.
While you're correct, your logic is flawed. Because by your argument, the function would not be well-defined over positive real numbers either. Consider how the square root of 4 could be 2 or -2 by your exact argument.
more importantly, the square of a general complex number is another complex number that is specifically not a negative number unless it is pure imaginary: (a+bi)^2 = a^2-b^2+2abi For that to be negative a or b must be zero. If a is zero the first number is in the subset of imaginary numbers anyway, or if b is zero then the square is positive and does not fit the clue.
please please allow 'complex' numbers. i know that if you want to get entirely technical then complex numbers have two parts but it's the general term that is now being taught without most teachers grasping the distinction. besides, the term 'imaginary' is far far worse at summing up the concept and we're trying to retire the phrase where possible